Numerical analysis, Mechanics
Shape optimization using a level set based mesh evolution method: an overview and tutorial
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1267-1332.

This article revolves around a recent numerical framework for shape and topology optimization, which features an exact mesh of the shape at each iteration of the process, while still leaving the room for an arbitrary evolution of the latter (including changes in its topology). In a nutshell, two complementary representations of the shape are combined: on the one hand, it is meshed exactly, which allows for precise mechanical calculations based on the finite element method; on the other hand, it is described implicitly, using the level set method, which makes it possible to track its evolution in a robust way. In the first part of this work, we overview the main aspects of this numerical strategy. After a brief presentation of some necessary background material – related to shape optimization and meshing, among others – we describe the numerical schemes involved, notably when it comes to the practice of the level set method, the remeshing algorithms, and the considered optimization solver. This strategy is illustrated with 2d and 3d numerical examples in various physical contexts. In the second part of this article, we propose a simple albeit efficient python-based implementation of this framework. The code is described with a fair amount of details, and it is expected that the reader can easily elaborate upon the presented examples to tackle his own problems.

Cet article traite d’un cadre de travail récent dédié à la résolution numérique de problèmes d’optimisation de formes  ; il s’illustre par une représentation exacte, maillée, de la forme à chaque itération du procédé, tout en laissant la place à une évolution arbitraire de celle-ci (y compris des changements de sa topologie). L’idée centrale de cette stratégie est de combiner deux représentations complémentaires de la forme  : d’une part, celle-ci est maillée explicitement, de sorte qu’il est possible d’effectuer des calculs mécaniques précis par la méthode des éléments finis  ; d’autre part, elle est décrite implicitement, par la méthode des lignes de niveaux, facilitant ainsi le suivi robuste de son évolution. Dans la première partie de ce travail, on résume les points saillants de cette stratégie numérique. Après avoir brièvement rappelé quelques notions de base – en lien, entre autres, avec l’optimisation de formes et le maillage – on décrit les schémas numériques mis en jeu, notamment pour la pratique de la méthode des lignes de niveaux, les algorithmes de remaillage, et l’algorithme d’optimisation numérique. Cette méthodologie est illustrée par des exemples numériques en deux et trois dimensions d’espace, dans différents contextes physiques. Dans la seconde partie de cet article, on propose une implémentation python open-source, simple mais efficace, de ce cadre de travail. Le code est détaillé de sorte que le lecteur puisse facilement intervenir dedans et le modifier pour traiter un problème de son choix.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.498
Classification: 49M41, 65K05, 65N50, 74P05, 74P10, 90C90

Charles Dapogny 1; Florian Feppon 2

1 Univ. Grenoble Alpes, CNRS, Grenoble INP (Institute of Engineering Univ. Grenoble Alpes), LJK, 38000 Grenoble, France
2 Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Heverlee, Belgium
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Charles Dapogny; Florian Feppon. Shape optimization using a level set based mesh evolution method: an overview and tutorial. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1267-1332. doi : 10.5802/crmath.498. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.498/

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